# How to derive a formula for the period of a simple pendulum? [duplicate]

The following formula is given in our lab manual: $$T = 2 \pi \sqrt{\frac{L}{g}} \left( 1 + \frac{1}{4}\sin^2 \frac{\theta}{2} + \frac{9}{64}\sin^4 \frac{\theta}{2}+\cdots \right)$$ for the period of a simple pendulum with length $$L$$ under the influence of gravitational acceleration with magnitude $$g$$. We ignore friction. I understand the solution of Newton's Equation via the small angle approximation $$\sin \theta \approxeq \theta$$ and why that gives the usual elementary formula $$T = 2 \pi \sqrt{\frac{L}{g}}$$. What I don't understand is how one derives the above formula for higher-order corrections to the period. I think the $$\theta$$ in the formula is the maximum angle to which the pendulum in question is swinging.

Any insight into the origin or derivation of this formula is appreciated.

## marked as duplicate by Bill N, John Rennie newtonian-mechanics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jan 23 at 6:16

• See the section “Legendre polynomial solution for the elliptic integral” at en.wikipedia.org/wiki/… – G. Smith Jan 22 at 17:24
• BTW, there's a much nicer formula if you're doing this on a computer, using the AGM, the arithmetic-geometric mean, which converges very quickly. – PM 2Ring Jan 22 at 17:31
• Possible duplicate of Period of a pendulum See the answer here. The solution is found in different textbooks, too. e.g., Engineering Dynamics by Beer & Johnston, , Classical Dynamics by Marion, This derivation is also a problem in Classical Mchanics by Taylor, so OP might be looking for that .... – Bill N Jan 22 at 18:30
• That answer doesn’t explain how to get the series. – G. Smith Jan 22 at 18:43
• @BillN I don't spend much time over here in Physics, but that seems pretty far from a direct duplicate. If I was searching for the series formula for the period I don't think that Q and A would be directly helpful. – James S. Cook Jan 23 at 1:14

If we take the zero of gravitational potential energy to be at the pivot, then by conservation of energy,

$$\frac{1}{2}m(L\dot{\theta})^2-m g L\cos{\theta}=-m g L\cos{\theta_0}$$

where $$\theta_0$$ is the maximum angle (where the kinetic energy is zero). This gives the differential equation

$$\frac{d\theta}{dt}=\sqrt{\frac{2g}{L}(\cos{\theta}-\cos{\theta_0}})$$

which gives the period as four times the time to go from 0 to $$\theta_0$$:

$$T=4\sqrt\frac{L}{2g}\int_0^{\theta_0}\frac{d\theta}{\sqrt{\cos{\theta}-\cos{\theta_0}}}.$$

Using the trigonometric identity

$$\cos\theta=1-2\sin^2{\frac{\theta}{2}}$$

this becomes

$$T=2\sqrt\frac{L}{g}\int_0^{\theta_0}\frac{d\theta}{\sqrt{\sin^2{\frac{\theta_0}{2}}-\sin^2{\frac{\theta}{2}}}}.$$

With the change of integration variable to $$u$$ where

$$\sin{u}=\frac{\sin{\frac{\theta}{2}}}{\sin{\frac{\theta_0}{2}}}$$

this becomes

$$T=4\sqrt\frac{L}{g}\int_0^{\pi/2}\frac{du}{\sqrt{1-k^2\sin^2{u}}}$$

where

$$k=\sin\frac{\theta_0}{2}.$$

Now use a Taylor expansion and integrate term-by-term to get the result you were given:

\begin{align} T&=4\sqrt\frac{L}{g}\int_0^{\pi/2}du\;\left(1+\frac{1}{2}k^2\sin^2{u}+\frac{3}{8}k^4\sin^4{u}+...\right)\\ &=4\sqrt\frac{L}{g}\left(\frac{\pi}{2}+\frac{1}{2}k^2\frac{\pi}{4}+\frac{3}{8}k^4\frac{3\pi}{16}+...\right) \\ &=2\pi\sqrt\frac{L}{g}\left(1+\frac{1}{4}k^2+\frac{9}{64}k^4+...\right). \end{align}

• I think you'd enjoy The AGM Simple Pendulum by Mark B. Villarino. – PM 2Ring Jan 22 at 18:45
• Thanks for this answer. Exactly the sort of detail I had hoped for. – James S. Cook Jan 22 at 21:21